Apologies for the long silence on this blog! I’ve been distracted for the past month with other things. I’m taking several interesting math classes this semester, one of which I have been liveTeXing and one of which has (at least for now) been providing notes. I’ve also been taking the Kan seminar at MIT. I recently gave a half-hour talk for that class, on Segal’s paper “Categories and cohomology theories;” as a short talk, it focused mostly on the key definitions in the paper rather than the proofs. (Reading Gian-Carlo Rota’s book has convinced me that’s not a bad thing.)  This post contains the notes for that talk.

The goal of this talk is to motivate the notion of “homotopy coherence,” and in particular the example of (special) ${\Gamma}$-spaces. In particular, the goal is to find a “homotopy coherent” substitute for the notion of a topological abelian monoid.

Why might we want such a notion? Given a topological abelian monoid ${A}$, one can form the classifying space ${BA}$, which still acquires the structure of a topological abelian monoid (if we use the usual simplicial construction). It follows that we can iterate the construction, producing a sequence of topological abelian monoids

$\displaystyle A, BA, B^2 A, \dots \ \ \ \ \ (1)$

together with maps ${B^{n-1}A \rightarrow \Omega B^nA}$. If ${A}$ is a topological abelian group, then these maps are all equivalences, and we have an infinite delooping of ${A}$. Therefore, we can extract a cohomology theory from ${[\cdot, A]}$.

If ${A}$ is not assumed to be a group, then the maps ${B^{n-1} A \rightarrow \Omega B^n A}$ are still equivalences for ${n \geq 2}$ and we get a cohomology theory out of ${[\cdot, \Omega BA]}$. This ability to extract an infinite loop space is a very desirable property of topological abelian monoids. Unfortunately, topological abelian monoids are always products of Eilenberg-MacLane spaces, but we’d like to deloop other spaces. Motivated by this, let us take the delooping question as central and declare:

Requirement: A good definition of a “homotopy coherent” topological abelian monoid should have the following properties. If ${X}$ is a homotopy coherent topological monoid, then:

1. ${X}$ should come with the structure of a homotopy commutative ${H}$ space.
2. We should be able to build a sequence of spaces ${X, BX, B^2 X, \dots }$ and maps ${B^{n-1 } X \rightarrow \Omega B^n X}$ such that if ${X}$ is grouplike (i.e., ${\pi_0 X}$ is a group), then these maps are equivalences (so that we get an infinite delooping of ${X}$).
3. Conversely, any infinite loop space should be a candidate for ${X}$.
4. Finally, the definition should not require explicitly assuming a delooping to begin with!

Example 1 The space ${BU \times \mathbb{Z}}$ is a homotopy commutative ${H}$ space (classifying stable vector bundles). It is not a topological abelian group (otherwise, ${K}$-theory would be periodic integral cohomology!), but we want ${BU \times \mathbb{Z}}$ to be an example of a “homotopy coherent” abelian group as ${BU \times \mathbb{Z}}$ classifies stable vector bundles, which come with an addition which we would like to believe is “coherently” commutative. In particular, ${BU \times \mathbb{Z}}$ is an infinite loop space via Bott periodicity ${BU \times \mathbb{Z} \simeq \Omega^2( BU \times \mathbb{Z})}$.

We might start by suggesting a commutative monoid in the homotopy category of spaces: that is, a homotopy commutative H space. In other words, it is a space ${X}$ together with a multiplication ${m: X \times X}$ and a unit ${e: \ast \rightarrow X}$ which satisfy the axioms of a commutative monoid, up to homotopy. Let’s start with this, and see why it is a bad definition.

Bad definition: A “homotopy coherent” topological abelian monoid is a homotopy commutative (and associative) H space.

This is not an adequate definition. For one thing, the data of a homotopy commutative H space is insufficient to run the Milnor construction and to build the classifying space (let alone the iterated classifying spaces).

Example 2 Let ${n \geq 3}$. It is a theorem of Adams (in his paper “The sphere considered as an H space mod $p$“) that the localization ${S^{2n-1}_{(p)}}$ is a homotopy associative, homotopy commutative H space for ${p > 3}$, but Adams also shows that for infinitely many ${p}$, a topological associative monoid with that mod ${p}$ homology cannot exist.

A homotopy commutative H space is a space together with a multiplication which is homotopy commutative and associative. In the homotopy coherent setting, we want to keep track of the homotopies which verify commutativity (or associativity) of the product, and we want these homotopies to satisfy higher coherence conditions (which are verified by certain even higher homotopies, which satisfy their own coherence conditions).

Analogy: Recall the notion of a symmetric monoidal category. A symmetric monoidal category ${(\mathcal{C}, \otimes)}$ comes with a bifunctor ${\otimes: \mathcal{C} \times \mathcal{C} \rightarrow \mathcal{C}}$ and natural isomorphisms (“homotopies”) describing commutativity and associativity of ${\otimes}$, which are part of the data. These relations are required to satisfy coherence conditions (if we were working with higher categories we would have higher coherence homotopies). Segal shows that the notion of a special ${\Gamma}$-space satisfies the above requirements, and exhibits infinite loop structures on a variety of spaces of interest. It turns out that one can still extract from a special ${\Gamma}$-space a sequence as in (1).

To motivate the definition of a special ${\Gamma}$-space, let’s recall that an abelian monoid in a category ${\mathcal{C}}$ with products is an object ${A}$ together with morphisms ${e: \ast \rightarrow A}$ and ${m : A \times A \rightarrow A}$ satisfying various relations. It is awkward to say “satisfying various relations” in a homotopy-theoretic setting. Roughly speaking, for every relation (such as the relation that ${m: A \times A \rightarrow A}$ is equal to ${m \circ \mathrm{flip}: A \times A\rightarrow A}$) we want a homotopy verifying this relation, together with coherence data on these homotopies. Unfortunately, it is awkward to spell out all these coherence homotopies in detail: there are infinitely many of them!

Fortunately, there are other ways of phrasing the definition of an abelian monoid, which do not require explicit relations. Segal uses a “diagrammatic” version of the definition which generalizes naturally to the homotopy setting.

Definition 1 Let ${\Gamma^{op}}$ be the category of finite sets and partially defined maps. (Given partially defined maps ${f: S \rightarrow T}$ and ${g: T \rightarrow R}$, then the composite morphism ${S \rightarrow T \rightarrow R}$ is defined on ${f^{-1}(\mathrm{dom}(g)) \circ \mathrm{dom} f}$.)

Let ${A}$ be an abelian monoid. Then we get a functor ${F_A: \Gamma^{op} \rightarrow \mathbf{Sets}}$ defined as follows:

1. If ${S}$ is a finite set, then ${F_A(S) = A^S}$.
2. If ${f: S \rightarrow T}$ is a partially defined map, then we define ${F_A(f): A^S \rightarrow A^T}$ by

$\displaystyle F_A(f)(\left\{a_s\right\})_t = \sum_{s' \in f^{-1}(t)} a_{s'}.$

The above data encodes the entire structure of abelian monoid on ${A}$. In fact, we can go in the other direction, as well, under certain conditions.

Proposition 2 Any functor ${F: \Gamma^{op} \rightarrow \mathbf{Sets}}$ such that for any finite sets ${S, T}$, the map ${F( S \sqcup T) \rightarrow F(S) \times F(T)}$ (arising from the partially defined maps ${S \sqcup T \rightarrow S, S \sqcup T \rightarrow T}$) is an isomorphism and ${F(\emptyset) = \ast}$, arises as ${F_A}$ for a unique abelian monoid ${A}$.

In fact, we have an equivalence between abelian monoids and functors ${F: \Gamma^{op} \rightarrow \mathbf{Sets}}$ which turn coproducts into products. This is true in any category ${\mathcal{C}}$ with finite products. A functor ${F: \Gamma^{op} \rightarrow \mathcal{C}}$ that turns coproducts into products is the same thing as an abelian monoid object in ${\mathcal{C}}$. For instance, if ${\mathcal{C} = \mathbf{Spaces}}$ is the category of topological spaces, then a functor ${F: \Gamma^{op} \rightarrow \mathbf{Spaces}}$ which turns coproducts into products is precisely a topological abelian monoid. We thus get a “diagrammatic” reformulation of the axioms defining a topological abelian monoid.

Definition 3 (Segal) A special ${\Gamma}$-space is a functor ${F: \Gamma^{op} \rightarrow \mathbf{Spaces}}$ with the property that for finite sets ${S, T}$, the map

$\displaystyle F(S \sqcup T) \rightarrow F(S) \times F(T)$

is a homotopy equivalence.

In particular, the space ${F(\ast)}$ acquires the structure of a homotopy commutative H space, as the functor ${F: \Gamma^{op} \rightarrow \mathbf{Spaces} \rightarrow \mathrm{Ho}(\mathbf{Spaces})}$ is product-preserving. Observe that a ${\Gamma}$-space ${F}$ is in some sense a type of structure placed on the space ${F(\ast)}$, and we will abuse notation and talk about “${\Gamma}$-structures” on a space ${T}$ (meaning a ${\Gamma}$-space extending ${T}$).

Example 3 Not every product-preserving functor ${F: \Gamma^{op} \rightarrow \mathrm{Ho}(\mathbf{Spaces})}$ lifts to a functor ${\Gamma^{op} \rightarrow \mathbf{Spaces}}$: that is, there are homotopy commutative H spaces which are not ${\Gamma}$-spaces, since not every homotopy commutative groulike H space can be delooped. This provides examples of homotopy commutative diagrams of spaces which cannot be strictified.

Thesis:  ${\Gamma}$-spaces are the “correct” homotopy coherent analog of an topological abelian monoid.

One way to “prove” this thesis is to show that ${\Gamma}$-spaces satisfy the requirements. For instance, we should be able to place ${\Gamma}$-structures on numerous spaces (such as ${BU \times \mathbb{Z}}$), and on the other hand we should have enough structure to perform operations such as forming the classifying space and group completion. The following result shows that there is a rich supply of ${\Gamma}$-spaces arising from symmetric monoidal categories.

Theorem 4 (Segal) Let ${\mathcal{C}}$ be a symmetric monoidal category. Then the nerve (recall that ${\iota}$ means to take the subcategory of isomorphisms) ${N(\iota \mathcal{C})}$ canonically admits the structure of a ${\Gamma}$-space.

The intuition behind this result is that a $\Gamma$-category is really the same thing as a symmetric monoidal category; the notion of a symmetric monoidal category answers the question at the beginning of this post if we were working with categories rather than spaces.

Example 4 Consider the category of finite sets, which is a symmetric monoidal category under the coproduct. The nerve is ${\bigsqcup B \Sigma_n}$, which is consequently a ${\Gamma}$-space (it is essentially the “free” ${\Gamma}$-space on a point, i.e. it is the “free homotopy algebra” ${\bigsqcup (\ast^n)_{h \Sigma_n}}$). In other words, given a ${\Gamma}$-structure on a space ${X}$, to give a map of ${\Gamma}$-spaces ${\bigsqcup B \Sigma_n \rightarrow X}$ up to homotopy is equivalent to giving an element of ${\pi_0 X}$.

Observe that the composition structure on ${\bigsqcup B \Sigma_n}$ comes from juxtaposition of permutations ${\Sigma_n \times \Sigma_m \rightarrow \Sigma_{n+m}}$. In fact, ${\bigsqcup B \Sigma_n}$ is a strictly associative topological monoid.

Example 5 Let’s now see why ${BU \times \mathbb{Z}}$ is a ${\Gamma}$-space, or at least half of the reason. Consider thetopological category of finite-dimensional ${\mathbb{C}}$-vector spaces and isomorphisms. This is a symmetric monoidal (topologically enriched) category under the direct sum. We can make sense of the nerve of a topologically enriched category (it is a geometric realization of a simplicial spacenow), and the above theorem makes sense for such. Observe that we have an equivalence

$\displaystyle N(\mathcal{C}) \simeq \bigsqcup BU(n).$

According to the above result, ${\bigsqcup BU(n)}$ is a ${\Gamma}$-space. One can show from this that ${BU \times \mathbb{Z}}$ is a ${\Gamma}$-space using the “group completion” theorem.

Finally, let’s sketch the delooping construction. Given a ${\Gamma}$-space ${X}$, we would like to extract a sequence of spaces

$\displaystyle X, BX, B^2X , \dots$

together with maps ${B^n X \rightarrow \Omega B^{n+1} X}$ which are equivalences for ${n \geq 1}$. If ${\pi_0 X}$ is a group, then ${X \rightarrow \Omega BX}$ is an equivalence as well, and we get on ${X}$ the structure of aninfinite loop space. In order to do this, recall that the classical construction of ${BG}$ for a topological group ${G}$ is the geometric realization of a simplicial space which looks like ${\ast, G, G \times G, \dots }$. Segal’s observation is that a ${\Gamma}$-space provides enough structure to build such a simplicial space, and thus a geometric realization.

Unfortunately, given a ${\Gamma}$-space ${X: \Gamma^{op} \rightarrow \mathbf{Spaces}}$, we cannot build a simplicial space from ${X \times X \stackrel{\stackrel{\rightarrow}{\rightarrow}}{\rightarrow} X \rightrightarrows \ast}$: the simplicial identities will not be verified. However, we don’t really need ${X \times X}$ there; we could replace it with something equivalent to it. Segal produces a simplicial space which looks like

$\displaystyle \dots X( \left\{1,2 \right\}) \stackrel{\stackrel{\rightarrow}{\rightarrow}}{\rightarrow} X(\left\{1\right\}) \rightrightarrows X(\ast),$

where the maps are defined using the combinatorics of ${\Gamma}$.

Definition 5 We define a functor ${\Delta^{op} \rightarrow \Gamma^{op}}$ as follows. Given a finite nonempty totally ordered set ${A}$, we send it to ${A \setminus \left\{0\right\} }$ where ${0 \in A}$ is the smallest element. Given a map ${f: A \rightarrow B}$ of totally ordered sets, we have to define a partially defined map of sets ${B \setminus \left\{0_B\right\} \rightarrow A \setminus \left\{0_A\right\}}$. We send ${b \in B \setminus \left\{0\right\}}$ to ${\min_{a \in A \setminus \left\{0\right\}, b\leq f(a) } a}$ (which is undefined if no such ${a}$ exist).

Definition 6 Given a ${\Gamma}$-space ${X}$, we let ${BX}$ be the geometric realization of the associated simplicial space. In fact, ${BX}$ itself acquires the structure of a ${\Gamma}$-space, assigning to a finite set ${S}$ the geometric realization of the ${\Gamma}$-space ${T \mapsto X(S \times T)}$. We can thus form ${B^2 X}$ and so forth.

The main result is:

Theorem 7 (Segal) If ${X}$ is grouplike, then ${B^n X \simeq \Omega B^{n+1}X}$ for all ${n \geq 0}$ and ${X}$ is an infinite loop space. In any event, we have ${B^n X \simeq \Omega B^{n+1} X}$ for ${n \geq 2}$.

What I didn’t have time for in the talk is the stronger result that special $\Gamma$-spaces which are grouplike are equivalent (as an $\infty$-category) to the $\infty$-category $\mathbf{Sp}_{\geq 0}$ of connective spectra. I’m not sure I yet have a really convincing reason in my mind for why this should be. One possible reason is that there is an equivalence between spectra and excisive functors from finite pointed spaces to spaces (a spectra $X$ defines an excisive functor $T \mapsto \Omega^{\infty} ( X \wedge \Sigma^{\infty} T)$), whereas special $\Gamma$-spaces are precisely excisive functors on finite discrete spaces. There has to be a result describing when left Kan extensions of excisive functors are excisive, but I don’t know it. Perhaps someone can help here?